Problem: Is ${77319}$ divisible by $3$ ?
Solution: A number is divisible by $3$ if the sum of its digits is divisible by $3$ . [ Why? First, we can break the number up by place value: $ \begin{eqnarray} {77319}= &&{7}\cdot10000+ \\&&{7}\cdot1000+ \\&&{3}\cdot100+ \\&&{1}\cdot10+ \\&&{9}\cdot1 \end{eqnarray} $ Next, we can rewrite each of the place values as $1$ plus a bunch of $9$ s: $ \begin{eqnarray} {77319}= &&{7}(9999+1)+ \\&&{7}(999+1)+ \\&&{3}(99+1)+ \\&&{1}(9+1)+ \\&&{9} \end{eqnarray} $ Now if we distribute and rearrange, we get this: $ \begin{eqnarray} {77319}= &&\gray{7\cdot9999}+ \\&&\gray{7\cdot999}+ \\&&\gray{3\cdot99}+ \\&&\gray{1\cdot9}+ \\&& {7}+{7}+{3}+{1}+{9} \end{eqnarray} $ Any number consisting only of $9$ s is a multiple of $3$ , so the first four terms must all be multiples of $3$ That means that to figure out whether the original number is divisible by $3 $ , all we need to do is add up the digits and see if the sum is divisible by $3$ . In other words, ${77319}$ is divisible by $3$ if ${ 7}+{7}+{3}+{1}+{9}$ is divisible by $3$ Add the digits of ${77319}$ $ {7}+{7}+{3}+{1}+{9} = {27} $ If ${27}$ is divisible by $3$ , then ${77319}$ must also be divisible by $3$ ${27}$ is divisible by $3$, therefore ${77319}$ must also be divisible by $3$.